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A251658
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a(n) = Sum_{k=1..n} C(n,k) * sigma(k)^(n-k) for n>=1.
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1
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1, 3, 13, 75, 471, 3273, 26916, 262027, 2851375, 33353433, 422098711, 5947420449, 94264842933, 1637995975164, 29950444958268, 558160521126939, 10427042476673645, 194220965933772135, 3611256888560663173, 67328916459842949045, 1266687562134302300334, 24238971105618591823339
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f.: Sum_{n>=1} exp(sigma(n)*x) * x^n/n!.
O.g.f.: Sum_{n>=1} x^n/(1 - sigma(n)*x)^(n+1).
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EXAMPLE
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E.g.f.: A(x) = x + 3*x^2/2! + 13*x^3/3! + 75*x^4/4! + 471*x^5/5! +...
where
A(x) = exp(x)*x + exp(3*x)*x^2/2! + exp(4*x)*x^3/3! + exp(7*x)*x^4/4! + exp(6*x)*x^5/5! + exp(12*x)*x^6/6! + exp(8*x)*x^7/7! + exp(15*x)*x^8/8! +...+ exp(sigma(n)*x)*x^n/n! +...
RELATED EXPRESSIONS.
O.g.f.: A(x) = x + 3*x^2 + 13*x^3 + 75*x^4 + 471*x^5 + 3273*x^6 +...
where
A(x) = x/(1-x)^2 + x^2/(1-3*x)^3 + x^3/(1-4*x)^4 + x^4/(1-7*x)^5 + x^5/(1-6*x)^6 + x^6/(1-12*x)^7 + x^7/(1-8*x)^8 + x^8/(1-15*x)^9 +...+ x^n/(1-sigma(n)*x)^(n+1) +...
L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 75*x^4/4 + 471*x^5/5 + 3273*x^6/6 + 26916*x^7/7 + 262027*x^8/8 + 2851375*x^9/9 +...
where exponentiation yields the g.f. of A291558:
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 123*x^5 + 695*x^6 + 4710*x^7 + 38573*x^8 + 363290*x^9 + 3762248*x^10 +...+ A291558(n)*x^n +...
which is an integer series.
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MATHEMATICA
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Table[Sum[Binomial[n, k] * DivisorSigma[1, k]^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 22 2017 *)
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PROG
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(PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(k)^(n-k))}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(1+sum(k=1, n, exp(sigma(k)*x +x*O(x^n))*x^k/k!), n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(k=1, n, x^k/(1-sigma(k)*x +x*O(x^n))^(k+1)), n)}
for(n=1, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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